I find this topic really fascinating and am intrigued by the perspectives everyone has contributed here (know I'm jumping in late...). I've noticed people generally approach this with a goal of how to 'get around' the tension or escape inciting any offense. It's as if science and faith are mutually exclusive, a sleeping dragon to be jumped over without getting bitten. And sure, always good to avoid getting bitten (or biting!)...but, particularly when it comes to medicine, why not look for the reconciliation point? View it as a quality of a person like any other, a piece of information that could be of potential benefit?
If someone needed exercise and you learned that they hated going to the gym or playing sports, but loved dogs, you might suggest they offer to walk a friend's dog daily. Presumably the whole mindset of an effective doctor is to cure or help to the best of his ability his patient. Whatever you personally believe, surely it can't hurt to try everything possible to get them better, provided it wouldn't harm them (and is within doctor's scope, legality, blahblah). I absolutely believe in willpower and there's evidence for the placebo effect, along with ongoing studies into mind-body connection (look at studies on depression, 'practicing' smiling in order to elicit a feedback loop).
If someone's beliefs were to come up, I think it's worth looking for ways to run with that instead of away from it. The same way you might (subconsciously) influence a parent to fight to live by talking to them about their children, if it helps a patient to believe a deity is watching over them, why not encourage them to hold onto their faith, whether by letting them talk about it or whatever, in hopes it'll help them recover? It shouldn't even require disclosing your own beliefs: is it possible to redirect? "What do you, the patient, believe? That's what matters here, not my beliefs."
As a general response to some previous points: if someone believes in God and thus His role in creating everything, that faith carries over to their 'belief' in medicine. If you believe God put you on this earth for some purpose, doesn't it follow he did the same with your physician? God's a busy guy; he needs to delegate tasks as much as any company boss
(which, to some extent, is why it confuses me when people denounce vaccines/drugs/inventions under the claim that "God or prayer alone will fix me"...it reminds me of that old joke:
A man gets hit by a car and prays to God to save him. An ambulance comes by and he sends it away saying, "God will save me." Another ambulance comes by and, though in bad shape, he holds fast and tells it to leave stating, "God will save me." But he dies. When he gets to the gates of Heaven and asks God, "God, why didn't you come to save me?", God replies, "I sent you two ambulances!")
But anyway, the point of all that being, rather than view faith-friction as an obstacle, why not look for ways to use a patient's worldview to help him/her without having to be untrue to yourself? You can't influence someone's actions (i.e. health, here...) by appealing to a system they don't believe in; this is a place where objective empathy (if that makes sense) rules all. I'm by no means saying this is necessary all the time, but I think looking for paths of benefit is a perspective ultimately worth more thought than being on the defensive.